Definition 4

Let $\alpha \mapsto f_\alpha$ be an action of a set $\Omega$ on a set $E$. An equivalence relation $R$ on $E$ is said to be compatible with the given action if, for all elements $x$ and $y$ of $E$ such that $x \equiv y \pmod{R}$ and all $\alpha \in \Omega, f_\alpha(x) \equiv f_\alpha(y) \pmod{R}$. The mapping which associates with an element $\alpha \in \Omega$ the mapping of $E/R$ into itself derived from $f_\alpha$ by passing to the quotients is an action of $\Omega$ on $E/R$ called the quotient of the action of $\Omega$ on $E$.

Let $E$ be a magma and $R$ an equivalence relation on $E$. $R$ is said to be left (resp. right) compatible with the law on $E$ if it is compatible with the left (resp.

right) action of E on itself derived from the law on E. For R to be compatible with the law on E it is necessary and sufficient that it be left and right compatible with the law on E.

We leave to the reader the statement and proof of the analogues of Propositions 6, 7 and 8 of § 1, no. 6.